IIT / JEE ADVANCED – VERY TOUGH PROBLEMS
Oscillations & Simple Harmonic Motion (SET–3)
Problem 1: SHM with Accelerating Frame (Elite Level)
A block of mass m attached to a spring (spring constant k) is kept on a smooth horizontal surface. The entire system is placed inside a lift accelerating upward with acceleration a. Find the time period of oscillation as observed from the lift.
Solution:
In non-inertial frame of lift, a pseudo force ma acts downward. This shifts the equilibrium position but does NOT change stiffness.
Restoring force remains proportional to displacement from new mean.
Therefore:
T = 2π √(m / k)
Answer: T = 2π √(m / k)
Topper Insight: Acceleration of frame shifts mean position, not time period.
Problem 2: SHM with Changing Mass (Advanced Concept)
A spring–mass system oscillates horizontally. Sand leaks slowly from the mass at constant rate. Assuming oscillations remain small, how does time period change?
Solution:
Time period:
T = 2π √(m / k)
As mass decreases gradually, time period decreases gradually.
Answer: Time period decreases with time
Exam Trap: Many students wrongly assume T remains constant.
Problem 3: Energy Distribution at Arbitrary Phase
In SHM, find the ratio of kinetic energy to potential energy when the phase angle is π/6.
Solution:
x = A cosθ θ = π/6 ⇒ cosθ = √3/2
PE / E = x² / A² = 3/4 ⇒ KE / E = 1/4
Therefore:
KE : PE = 1 : 3
Answer: 1 : 3
Problem 4: Two Connected Oscillators (Advanced)
Two identical masses connected by identical springs are placed on a smooth surface. If one mass is displaced slightly and released, find the nature of motion of the system.
Solution:
The system executes oscillations with two normal modes:
- In-phase motion
- Out-of-phase motion
Resultant motion is superposition of two SHMs with different frequencies.
Answer: Motion is periodic but not simple SHM
IIT Key Idea: More than one natural frequency ⇒ not SHM.
Problem 5: Force Law Identification
Which force law produces SHM for small oscillations about equilibrium?
- A) F = −kx
- B) F = −k sin x
- C) F = −kx²
- D) F = −k/x²
Correct Answers: A and B
Because for small x, sin x ≈ x.
Problem 6: Time Calculation Without Equation
Find the time taken for a particle in SHM to move from x = A/√2 to x = −A/√2.
Solution:
x = A cosθ A/√2 ⇒ θ₁ = π/4 −A/√2 ⇒ θ₂ = 3π/4
Δθ = π/2
Δt = (T / 2π) × (π / 2) = T / 4
Answer: T / 4
SET–3 MASTER TAKEAWAYS
- Acceleration changes mean, not frequency
- Changing mass changes time period
- Phase method dominates time problems
- Multiple frequencies destroy SHM
- Small-angle approximation is critical
IIT / JEE Tough Problems – SET–3 Completed ✅
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🔹 Simple Harmonic Motion (SHM) — Core Series
Coverage: Concepts → PYQs → Advanced Thinking → Exam Readiness
- SHM – Final Exam Day Checklist
- SHM Part 2
- SHM Part 3
- SHM Part 4
- SHM Part 5
- SHM Part 6
- SHM Part 7
- SHM Part 8
- SHM Part 9
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- SHM Part 11
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- SHM Part 15
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- SHM Part 17
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- SHM Part 21
- SHM Part 22
- SHM Part 23
- SHM Part 24
- SHM Part 25
- SHM Part 26
- SHM Part 27
- SHM Part 28
- SHM Part 29
🔹 Simple Harmonic Motion (SHM) — Extended Series (30–56)
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